Geometric Continuity of Parametric Curves

http://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/CSD-84-205.pdf

Parametric spline curves are typically constructed so that the first n parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as C^n or nth order parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves.

We define nth order geometric continuity (G^n), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. G^n continuity provides for the introduction of n quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.

BibTeX citation:

```@techreport{Barsky:CSD-84-205,
Author = {Barsky, Brian A. and DeRose, Anthony D.},
Title = {Geometric Continuity of Parametric Curves},
Institution = {EECS Department, University of California, Berkeley},
Year = {1984},
Month = {Oct},
URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/5752.html},
Number = {UCB/CSD-84-205},
Abstract = {Parametric spline curves are typically constructed so that the first <i>n</i> parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as <i>C^n</i> or <i>n</i>th order parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves.  <p>  We define <i>n</i>th order geometric continuity (<i>G^n</i>), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. <i>G^n</i> continuity provides for the introduction of <i>n</i> quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices.  Several applications of the theory are discussed, along with topics of future research.}
}
```

EndNote citation:

```%0 Report
%A Barsky, Brian A.
%A DeRose, Anthony D.
%T Geometric Continuity of Parametric Curves
%I EECS Department, University of California, Berkeley
%D 1984
%@ UCB/CSD-84-205
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/1984/5752.html
%F Barsky:CSD-84-205
```